Question

If a unit vector `veca` makes an angles `pi/3` with `hati, pi/4` with `hatj` and an acute angle θ with `hatk`, then find θ and, hence the compounds of `veca`.

Answer 1

`veca = a_1hati + a_2hatj + a_3hatk, |veca| = 1`

`cos  pi/3 = a_1/|veca|`

`1/2 = a_1[|veca| = 1]`

`cos  pi/4 = a_2/|veca|`

`1/sqrt2 = a_2[|veca| = 1]`

`cos theta = a_3/|veca|`

`a_3 = costheta`

|a| = 1

`sqrt(a_1^2 + a_2^2 + a_3^2) = 1`

`(1/2)^2 + (1/sqrt2)^2 + cos^2theta = 1`

`1/4 + 1/2 + cos^2theta = 1`

`cos^2theta = 1/4`

`costheta = 1/2`

`theta = pi/3`

`a_3 = cos  pi/3 = 1/2`

∴ `(1/2, 1/sqrt2, 1/2)`